I am revising and came over with this problem that I have no idea how to do!

Can somebody please help me?!

Consider arbitrary point x, y in the plane. It also has radial coordinate

r = (x^2 + y^2)^(1/2) and angular theta = arctan (y/x)

A small step is made in the vertical direction to a new point x, y +dx. Both r and theta change during the step so this corresponds to a small step dr radially, and rd(theta) in the angular direction.

How can i deduce dr and rd(theta) as functions of dy and theta (hint: what are the angles in the triangle formed, with side lengths rd(theta) and dr and hypotenuse dy?).

Hence deduce (∂r/∂y)_x as a function of theta.